K
INEMATICS
2.7
instantaneous acceleration is
a
0
. Find the average acceleration of the car over the 60° arc.(b)
The speed of an object undergoing uniform circular motion is 4
m/s
. The magnitude of the change in the velocity during 0.5 sec is also 4
m/s
. Find the minimum possible centripetal acceleration (in
m/s
2
) of the object. Q. 45.
A particle is fixed to the edge of a disk that is rotating uniformly in anticlockwise direction about its central axis. At time
t
= 0 the particle is on the
X
axis at the position shown in figure and it has velocity
v
y x v t
at = 0
(a) Draw a graph representing the variation of the
x
component of the velocity of the particle as a function of time. (b) Draw the
y
-component of the acceleration of the particle as a function of time.Q. 46. A disc is rotating with constant angular velocity
w
in anticlockwise direction. An insect sitting at the centre (which is origin of our co-ordinate system) begins to crawl along a radius at time
t
= 0 with a constant speed
V
relative to the disc. At time
t
= 0 the velocity of the insect is along the
X
direction. (a) Write the position vector
( )
of the insect at time ‘
t
’. (b) Write the velocity vector
( )
of the insect at time ‘
t
’. (c) Show that the
X
component of the velocity of the insect become zero when the disc has rotated through an angle
q
given by tan
q q
=
1.
O
v
w
X
y
Q. 47. (a) A point moving in a circle of radius
R
has a tangential component of acceleration that is always
n
times the normal component of acceleration (radial acceleration). At a certain instant speed of particle is
v
0
. What is its speed after completing one revolution?(b)The tangential acceleration of a particle moving in
xy
plane is given by
a
t
=
a
0
cos
q
. Where
a
0
is a positive constant and
q
is the angle that the velocity vector makes with the positive direction of
X
axis. Assuming the speed of the particle to be zero at
x
= 0, find the dependence of its speed on its
x
co-ordinate.Q. 48. A particle is rotating in a circle. When it is at point
A
its speed is
V
. The speed increases to 2
V
by the time the particle moves to
B
. Find the magnitude of change in velocity of the particle as it travels from
A
to
B
. Also, find
V V
A
D
; where
V
A
is its velocity at point A and
D
V
is change in velocity as it moves from
A
to
B
.Q. 49. A particle starts from rest moves on a circle with its speed increasing at a constant rate of . Find the angle through which it 0.8
ms
–2
would have turned by the time its acceleration becomes 1
ms
2
.Q. 50. In the arrangement shown in the fig, end
A
of the string is being pulled with a constant horizontal velocity of 6
m/s
. The block is free to slide on the horizontal surface and all string segments are horizontal. Find the velocity of point
P
on the thread.
A6 m/sP
Q. 51. In the arrangement shown in the fig, block
A
is pulled so that it moves horizontally along the line
AX
with constant velocity
u
. Block
B
moves along the incline. Find the time taken by
B
to reach the pulley
P
if
u
= 1
m/s
. The string is inextensible.
2 mA
u
PB
q
=30
0
1 2 m
X